Given a formal power series $f(x,y)=\sum_{n,m\ge 0}f(n,m)\: x^n y^m$ in two variables $x$ and $y$ over some field of characteristic zero, e.g. the field of complex numbers $\mathbb C$, we define a new formal power series in one variable $t$:

$\Delta(f)(t):=\sum_{i\ge 0} f(i,i) \:t^i$.

It is known that if $f(x,y)$ is rational $\Delta(f)(t)$ is in general not rational (I think it is algebraic though).

Example: $f(x,y)=\frac{1}{1-x-y}$ leads to $\Delta(f)(t)=(1-4t)^{-1/2}$.

The question is: Is there a constructive way (i.e. algorithm or explicit formula) to calculate $\Delta(f)(t)$ for rational (or algebraic) $f(x,y)$ ?